I am going to to take this thread at face value and in good faith.

So, lil zoomer boi, you got a PS2. One thing to keep in mind is, many games on the PS2 have been released in a decent enough state somewhere else or had vastly superior versions on Xbox.

So, keeping with that, and assuming you want to keep the console "stock", without any modding (meaning price would be a factor in "game pickups"). And you want to collect these cool little boxes, artifacts of gaming. Consider these games...

Ridge Racer V (Pricecharting CIB - $17 ish)
>Exclusive to PS2, can't play this anywhere else.

Really reflects the vibe of the console and the culture at the time. Easy to learn, hard to master, light on content, but what is there is very replayable.

SSX (Pricecharting CIB - $9 ish)
>Exclusive to PS2, can't play this anywhere else.

The first in the series of the iconic SSX snowboarding games. Super fun.

The Bouncer (Pricecharting CIB - $25 ish)
>Exclusive to PS2, can't play this anywhere else.
SquareSoft's experiment with a cinematic 3D Brawler. Got a lot of hate back in the day, but I think it's been re-appraised as something decent. Give a shot.

Final Fantasy VII Dirge of Cerberus (Pricecharting CIB - $22 ish)
>Exclusive to PS2, can't play this anywhere else.
A spinoff in the Final Fantasy VII cinematic universe or whatever that follows Vincent the vampire from the iconic RPG. A third person action shooting game+

Final Fantasy X (Pricecharting CIB - $12 ish)
>REMASTER AVAILABLE - but it's shit
One of the jest Jarpigs, imo.